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Application of the Phenomenological Renormalization to Percolation and Lattice Animals in Dimension 2

Bernard Derrida, Laurent de Seze

Published Jan 1, 1988• 9 citations
Publisher
DOI
10.1016/b978-0-444-87109-1.50024-8

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Principles and applications of a direct position space renormalization group for lattice models of polymers—called the Cluster Renormalization (CR)—are reviewed in detail. Particular emphasis is placed on the study of crossover phenomena and determination of universality classes in polymer models. The first part is largely a pedagogical description of the scaling concept and the idea of fractal dimensionality for polymers, and discusses the relation between polymer statistics and critical phenomena. The generalized lattice animal model for polymers is then introduced and it is shown that within a grand canonical ensemble polymers can be described as critical objects. This description enables us to apply CR to polymer models. Next, essentials of the CR are presented. Discussion of the applications begins with a single‐parameter CR study of the scaling properties of models of linear polymers, branched polymers and polymer networks. Finally various applications of two‐parameters CR to crossover phenomena are discussed. These include a study of the crossover from a random walk to a renormalization group approach for random walks, and a discussion of the effects of branches and loops on the universality classes of polymers. It also deals with the effects of screening in solutions of branched polymers and the crossover from lattice animals to percolation.
We study analytically the logarithmic corrections to the critical exponents of the critical behaviour of correlation length, susceptibility and specific heat for the temperature and the finite-size scaling behaviour, for a generic theory at its upper critical dimension (six). We have also computed the leading correction to scaling as a function of the lattice size. We distinguish the obtained formulae to the following special cases: percolation, Lee-Yang singularities and m -component spin glasses. We have compared our results for the Ising spin glass case with numerical simulations finding a very good agreement. Finally, and using the results obtained for the Lee-Yang singularities in six dimensions, we have computed the logarithmic corrections to the singular part of the free energy for lattice animals in eight dimensions.